Quasi-periodic Solutions for a Generalized Higher-Order Boussinesq Equation

Yanling Shi ^[1] ; Junxiang Xu ^[2]
1. [1] Yancheng Institute of Technology
  
  Yancheng Institute of Technology
  
  China
2. [2] Southeast University
  
  Southeast University
  
  China
Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 4, 2023
Idioma: inglés
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Resumen
- In this paper one-dimensional generalized eighth-order Boussinesq equation utt − ∂2 x u + β∂4 x u − ∂6 x u + ∂8 x u + (u3)x x = 0, β = ±1 with the boundary conditions u(0, t) = u(π, t) = ux x (0, t) = ux x (π, t) = uxxxx (0, t) = uxxxx (π, t) = 0 is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form.